Desargues's Theorem

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The following theorem of projective geometry:


Consider two triangles lying in the plane or positioned in 3-dimensional space. If the lines joining corresponding vertices are concurrent, then corresponding sides intersect in points that are collinear, and conversely.

In detail, suppose that one triangle has vertices A, B and C, and the other has vertices A′, B′ and C′. The theorem states that if AA′, BB′ and CC′ are concurrent, then BC and BC′ intersect at a point L, CA and CA′ intersect at M, and AB and AB′ intersect at N, where L, M and N are collinear.

Perhaps surprisingly, the case of three dimensions is the easier to prove. The theorem can be taken as a theorem of Euclidean geometry if suitable amendments are made to cover the possibility that points of intersection do not exist because lines are parallel. It played an important role in the emergence of projective geometry.

Subjects: Mathematics.

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